This version relies on paths rather than embeddings or induced subgraphs. I
much prefer it.
I am also optimistic that this progress will yield good Pi00 sentences. Say
tuned.
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Pi01 INCOMPLETENESS: digraph paths
by
Harvey M. Friedman
February 7, 2006
In this abstract, a digraph is a directed graph with no loops and no
multiple edges. Thus all digraphs will be simple. The results will be the
same if we allow loops.
A dag is a directed graph with no cycles.
Let G be a digraph. We write V(G) for the set of all vertices in G, and E(G)
for the set of all edges in G.
Let A containedin V(G). We write GA for the set of all destinations of edges
in G whose origins lie in A. I.e., GA = {y: (therexists x)((x,y) in E(G))}.
We begin by quoting a well known theorem about directed acyclic graphs, or
so called dags. We call it the complementation theorem, but we have been
told that it is rather ordinary fundamental fare in dag theory.
COMPLEMENTATION THEOREM (finite dags). Let G be a finite dag. There is a
unique set A containedin V(G) such that GA = V(G)\A.
We can look at the Complementation Theorem in terms of a large independent
set. We say that A containedin V(G) is independent in G if and only if there
is no edge connecting any two elements of A.
COMPLEMENTATION THEOREM (finite dags). Every finite dag has a unique
independent set A such that V(G)\A containedin GA.
A digraph on a set E is a digraph G where V(G) = E.
We will focus on digraphs whose vertex set is of the form [1,n]^k. Here
k,n >= 1 and [1,n] = {1,2,...,n}.
An upgraph on [1,n]^k is a digraph on [1,n]^k such that for all (x,y) in
E(G), max(x) < max(y).
The following is an immediate consequence of the Complementation Theorem
(finite dags) since upgraphs are obviously dags.
COMPLEMENTATION THEOREM (finite upgraphs). For all k,n >= 1, every upgraph
on [1,n]^k has a unique independent set A such that V(G)\A containedin GA.
Our development relies on what we call order invariant digraphs on [1,n]^k.
These are the digraphs G on [1,n]^k where only the relative order of
coordinates of pairs of vertices determine if they are connected by an edge.
More formally, let u,v lie in {1,2,3,...}^p. We say that u,v are order
equivalent if and only if for all 1 <= i,j <= p,
u_i < u_j iff v_i < v_j.
Let G be a digraph on [1,n]^k. We say that G is order invariant if and only
if the following holds. For all x,y,z,w in [1,n]^k, if (x,y) and (z,w) are
order equivalent (as 2k tuples), then
(x,y) in E(G) iff (z,w) in E(G).
Note that an order invariant digraph on [1,n]^k is completely determined,
among digraphs on [1,n]^k, by the subdigraph induced by [1,2k]^k -
regardless of how large n is. Thus the number of order invariant digraphs on
[1,n]^k is bounded by (2k)^k.
We write x! when x is a tuple of nonnegative integers. Here x! =
(x1!,...,xk!), where x has length k.
We say that
x starts a length r path in G continuing through S
if and only only if there exists x = x0,x1,...,xr such that each (xi,xi+1)
is an edge in G, and x1,...,xr lie in S.
PROPOSITION A. For all n,k,r >= 1, every order invariant upgraph G on
[1,n]^k has an independent set A such that every vertex x! that starts a
length r path in G continuing through V(G)\A, starts a length r path in G
continuing through GA in which the integer (8kr)!-1 does not appear.
Note that if we remove 'in which ...', then the statement immediately
follows from the Complementation Theorem (finite upgraphs), since we can use
the identity embeddings.
Proposition A can be proved with large cardinals but not without. Note that
Proposition A is explicitly Pi01.
Here is more detailed information.
Let MAH = ZFC + {there exists a strongly n-Mahlo cardinal}n.
Let MAH+ = ZFC + "for all n there exists a strongly n-Mahlo cardinal".
THEOREM 1. MAH+ proves Proposition A. However, Proposition A is not
provable in any consistent fragment of MAH that derives Z = Zermelo set
theory. In particular, Proposition A is not provable in ZFC, provided
ZFC is consistent. These facts are provable in RCA_0.
THEOREM 2. EFA + Con(MAH) proves Proposition A.
THEOREM 3. It is provable in ACA that Propositions A is equivalent to
Con(MAH).
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I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable
manuscripts. This is the 267th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-249 can be found at
http://www.cs.nyu.edu/pipermail/fom/2005-June/008999.html in the FOM
archives, 6/15/05, 9:18PM.
250. Extreme Cardinals/Pi01 7/31/05 8:34PM
251. Embedding Axioms 8/1/05 10:40AM
252. Pi01 Revisited 10/25/05 10:35PM
253. Pi01 Progress 10/26/05 6:32AM
254. Pi01 Progress/more 11/10/05 4:37AM
255. Controlling Pi01 11/12 5:10PM
256. NAME:finite inclusion theory 11/21/05 2:34AM
257. FIT/more 11/22/05 5:34AM
258. Pi01/Simplification/Restatement 11/27/05 2:12AM
259. Pi01 pointer 11/30/05 10:36AM
260. Pi01/simplification 12/3/05 3:11PM
261. Pi01/nicer 12/5/05 2:26AM
262. Correction/Restatement 12/9/05 10:13AM
263. Pi01/digraphs 1 1/13/06 1:11AM
264. Pi01/digraphs 2 1/27/06 11:34AM
265. Pi01/digraphs 2/more 1/28/06 2:46PM
266. Pi01/digraphs/unifying 2/4/06 5:27AM
Harvey Friedman